Directions in crystals

The disruption to the crystal introduced by a dislocation is characterized by the Burgers vector, b (see Supplementary Material SI for information on directions in crystals). During dislocation motion individual atoms move in a direction parallel to b, and the dislocation itself moves in a direction perpendicular to the dislocation line. As the energy of a dislocation is proportional to b2, dislocations with small Burgers vectors form more readily. 

The response of a crystal to an external stimulus such as a tensile stress, electric field, and so on is usually dependent upon the direction of the applied stimulus. It is therefore important to be able to specify directions in crystals in an unambiguous fashion. Directions are written generally as [uvw and are enclosed in square brackets. Note that the symbol [uvw] means all parallel directions or vectors. 

Point defects cannot be seen directly in crystals of flexible macromolecules, but have been deduced from calculations and models. The defect in Fig. 5.96 in the third chain from the left consists of a sequence of gauche, trans, and gauche conformations and is called a 2gl kink (see also Fig. 5.86). This disraption of the crystal is small, but it will be shown to assist in the motion of chains in aimealing and deformation, as well as contribute to the change in heat capacity beyond the vibrational contributions. 

Equations (17) and (18) further show a relationship between real and reciprocal space. The function F(hkl) is the Fourier transform of the unit cell contents, expressed in the reciprocal space coordinates h, k. and /. Because the. symmetry operation of translation holds for all three spatial directions in crystals, the Fourier transform of the entire crystal is zero, except at reciprocal lattice points. 

The connection between planes and the directions in crystals consists in the fact that the intersection of two planes is made after a line, which. 

Conversely, for the type I form of mxy8 stabilized upon miceUization, the N— C— C—X— C—X—N skeleton takes up an extended form, resulting in a densely packed state of n-octyl chains within the micelles. This dense packing results in high-order (crystalline) self-assembly. The molecular shape of pxy8 probably favors the formation of a densely packed aggregated state, resulting directly in crystal formation from the premicelles. 

It is possible to calculate derivatives of the free energy directly in a simulation, and thereby detennine free energy differences by thenuodynamic integration over a range of state points between die state of interest and one for which we know A exactly (the ideal gas, or hanuonic crystal for example) ... 

Under diffusion-controlled dissolution conditions (in the anodic direction) the crystal orientation has no influence on the reaction rate as only the mass transport conditions in the solution detennine the process. In other words, the material is removed unifonnly and electropolishing of the surface takes place. 

In the following development we consider a plane wave of infinite lateral extent traveling in the positive Xj direction (the wave front itself lies in the Xj, Xj plane). When discussing anisotropic materials we restrict discussion to those propagation directions which produce longitudinal particle motion only i.e., if u is the particle velocity, then Uj = Uj = 0. The <100>, <110>, and <111 > direction in cubic crystals have this property, for example. The derivations presented here are heuristic with emphasis on the essential qualitative features of plastic flow. References are provided for those interested in proper quantitative features of crystal anisotropy and nonlinear thermoelasticity. 

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